Power Functions - Algebra-2

1. Fundamental Concepts

Definition: Power functions are a type of polynomial function where the term with the highest degree is of the form $$ax^n$$, where $$a$$ is a constant and $$n$$ is a positive integer.
Graph Characteristics: The graph of a power function depends on the value of $$n$$. For even $$n$$, the graph resembles a parabola opening upwards or downwards. For odd $$n$$, the graph resembles a cubic function.
End Behavior: As $$x \rightarrow \infty$$ or $$x \rightarrow -\infty$$, the behavior of the function depends on the sign of $$a$$ and the parity of $$n$$.

Basic Rule: $$f(x) = ax^n$$

Degree Influence: The degree $$n$$ determines the shape and direction of the graph

Application: Power functions are used in physics to model phenomena such as gravitational force and electrical resistance

Problem: Identify the end behavior of the function $$f(x) = -2x^4 + 3x^2 - 5$$.

The leading term is $$-2x^4$$, which has an even degree and a negative coefficient.
For large values of $$x$$, the function behaves like $$-2x^4$$.
Since the degree is even and the coefficient is negative, the graph will open downwards.

Validation: As $$x \rightarrow \infty$$, $$f(x) \rightarrow -\infty$$; As $$x \rightarrow -\infty$$, $$f(x) \rightarrow -\infty$$ ✓

Problem: Graph the function $$g(x) = x^3 - 2x$$ and determine its zeros.

Identify the zeros by solving $$x^3 - 2x = 0$$.
Factor the equation: $$x(x^2 - 2) = 0$$.
Solve for $$x$$: $$x = 0$$, $$x = \sqrt{2}$$, $$x = -\sqrt{2}$$.
Plot these points and sketch the graph based on the odd degree and the sign changes around the zeros.

Validation: Zeros at $$x = 0$$, $$x = \sqrt{2}$$, $$x = -\sqrt{2}$$ ✓

Graphical Analysis: Use graphs to visualize the behavior of power functions and identify key features such as intercepts and end behavior.
Algebraic Manipulation: Factor polynomials to find zeros and understand the function's structure.
Pattern Recognition: Recognize patterns in the coefficients and degrees to predict the shape and orientation of the graph.